Vinayak Vatsal

Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k ≥ 2 and of the same level N, both eigenfunctions of the Hecke operators, and both normalized so that a₁ = 1. The main result we seek is that when E and f are congruent mod a prime p (which may be a prime ideal lying over a rational prime p > 2), the algebraic parts of the special values L(E,χ,j) and L(f,χ,j) satisfy congruences mod the same prime. More explicitly, the congruence result states that, under certain conditions,τ(χ ̄)L(f,χ,j)/(2πi)^(j−1)Ω_f^(sgn(E)) ≡ τ(χ ̄)L(E,χ,j)/(2πi)^(j)Ω_E (mod p)where the sign of E is ±1 depending on E, and Ω_f^(sgn(E)) is the correspondingcanonical period for f. Also, χ is a primitive Dirichlet character of conductor m, τ(χ ̄) is a Gauss sum, and j is an integer with 0
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This thesis consists of four chapters and deals with two different problems which are both related to the broad topic of special values of anticyclotomic L-functions. In Chapter 3, we generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic ℤ_p-extension. Let g be a cuspidal Hilbert modular form of parallel weight (2,...,2) and level N over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant D. We study the l-adic valuation of the special values L(g,χ,½) as χ varies over the ring class characters of K of P-power conductor, for some fixed prime ideal P. We prove our results under the only assumption that the prime to P part of N is relatively prime to D. In Chapter 4, we compute a basis for the two-dimensional subspace S_(k/₂)(Γ₀(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F ∈ S_(k₋₁)(Γ₀(N)), which satisfies L(F,½) ≠ 0. Here, we let k be a positive integer such that k ≡ 3 mod 4 and N be a positive square-free integer. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined. The squares of the Fourier coefficients of these forms are known to be essentially proportional to the central critical values of the L-function of F twisted by some quadratic characters.

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